The most funny parts to me in the movie is when the lady husband turn to an alien and how he acted to her "get your big butt back in the house". Then when he pulled back his face and then she fainted.
Next one is when Will tested at the center to be an MIB and then when Will had to use that little gun.
The part where Jay first meets Kay and they're in the car on the way to Jeebs' place and Jay asks him casually, 'so who exactly are you with? CIA? NSA?', at this point in the film being completely unaware of MIB and Kay just replies completely deadpan, 'I'm part of a special agency that polices and monitors alien activity on the planet Earth,' knowing that he won't be believed. Tommy Lee Jones ran away with this film in my opinion.
Also, 'no, ma'am, we at the FBI do not have a sense of humour we're aware of.' Genius delivery.
My favorite parts are J's incident with the tentacles in the car and the sound Frank makes when he is shaken and J explaining what it looks like to pedestrians. I have favorite parts from all 3 films so that's mine for the first.
When this movie was first publicized, I though, Oh, no, more aliens-among-us, gubbmint-conspiracy nonsense. Count me out. But then I saw a preview, and it was this scene that sold me. I still find it the most hilarious gag in the movie. Runner ups are: the "best of the best of the best, SIR!"; meeting Frank the Pug for the first time; Tony Shaloub, probably my favorite character actor, as Jeebs the pawnbroker; revealing Mikey the alien in the first scene; and interviewing Beatrice ("Oh, and, uh, hire a decorator to come in here real quick, cause... DAMN!")
I'm not sure it's the funniest part but when all the prospective recruits are taking the test got to me hard... by the time J is dragging the table I was actually crying of laughter in the cinema.